If we look into a dictionary or an encyclopedia, we see terms such as "Christianity," "Catholicism," "Protestantism," "Judaism," "Islam," "Hinduism," "Buddhism" all rather neatly defined and separate from eachother.
People like labels. They like nice clear concepts that they can move around and deploy like pieces on a chess-board.
It's likely that participants on a science-oriented discussion board feel that a lot more strongly than humanities-types. Tech-geeks are used to working with precisely-defined technical vocabularies, while poets love ambiguity, analogy and allusion.
But when we look into the world and the way one person who considers themselves a "Christian" differs from another who also considers themselves a "Christian," or a "Buddhist" from another "Buddhist" etc. etc. we see that these categories are not so clearly definable at all.
Many of our ordinary-language concepts are like that. They are what philosophers sometimes call 'family resemblance' concepts. The various examples embraced by a particular concept needn't necessarily share any single definitive essence in common. But they will share many characteristics in common with many other members of the set, even if there's no single characteristic that all of them possess.
How do we decide what is true "Christianity," true "Catholicism," true "Protestantism," true "Judaism," true "Islam," true "Hinduism," true "Buddhism"?
I'm inclined to perceive these things as historical traditions. So we would need to inquire into whether or not particular religious expressions are continuous with a tradition, are embraced by it or arise out of it.
Mahayana Buddhism differs from Pali Buddhism in various ways, sometimes so dramatically that they seem like different religions. But as an example, it's possible to show how the Boddhisattva ideal is present in embryonic form in the Pali suttas, how it grew in prominence in popular Buddhism and in the Jatakas, how it was shaped by the early controversites about Arahants' attainments, and so on.
So in Buddhism's case, we can watch the tradition getting broader and broader, as more and more ideas arose in it, eventually to the point where one side of the tradition didn't much resemble the other side. But there was still a sense that everyone was a Buddhist, a follower of the Buddha sasana, even if many Buddhists felt that people on the other extreme were adding uncanonical innovations or were practicing an inferior form of Buddhism. (It's not unlike Catholics and Protestants, who criticize each other but still accept each other as being Christians.)
Of course, in keeping with the family-resemblance nature of religious traditions, the historical continuity approach doesn't always work.
It's possible to make a very similar argument that Christianity and Judaism are continuous. The early new testament Christians certainly thought that they were. They were ethnic Jews, by and large, they believed and thought in thoroughly Jewish terms, they thought of the Hebrew scriptures as their own, and so on. There are no end of formative ideas of early Christianity in the inter-testamental Jewish apocalyptic literature. The new Christians were Jews preaching that the Messiah had come and that the Kingdom was dawning.
In this case, what made the two religions bifurcate into two distinct faiths is that most Jews didn't accepted the Christian claims, dismissed them as outlandish and impious, and unlike the Buddhist case, gradually came to perceive the new innovators as heretics and stopped accepting them as followers of their own Jewish faith. And within a few generations the new Christians were doing the same thing right back, rejecting the old Jews as outsiders, and imagining Christianity as if it was a whole new kind of Judaism that God had reached into history to create, with its own new covenant, destined to embrace the world and usher in the Kingdom.
